Will the inequality log M ≥ C*log 2 always hold, where M is a positive integer and C is the number of distinct prime numbers that divide M?
If so, prove it.
If not, give an example.
As C is positive and the log function is monotonically increasing, the inequation is equivalent to
M >= 2^C
which is true as each distinct prime divisor on the left is replaced by a 2, the smallest possible prime divisor, on the right. Not only is 2 less than or equal to every possible prime divisor on the left, but the "distinct" qualifier means there are usually more prime divisors on the left than on the right. In fact the only time the equality possibility holds is for M = 2.
3*5*7*7 >= 2*2*2
Posted by Charlie
on 2015-03-17 09:56:07